An important problem in network science is finding an optimal placement of sensors in nodes in order to uniquely detect failures in the network. This problem can be modelled as an identifying code set (ICS) problem, introduced by Karpovsky et al. in 1998. The ICS problem aims to find a cover of a set $S$, s.t. the elements in the cover define a unique signature for each of the elements of $S$, and to minimise the cover's cardinality. In this work, we study a generalised identifying code set (GICS) problem, where a unique signature must be found for each subset of $S$ that has a cardinality of at most $k$ (instead of just each element of $S$). The concept of an independent support of a Boolean formula was introduced by Chakraborty et al. in 2014 to speed up propositional model counting, by identifying a subset of variables whose truth assignments uniquely define those of the other variables. In this work, we introduce an extended version of independent support, grouped independent support (GIS), and show how to reduce the GICS problem to the GIS problem. We then propose a new solving method for finding a GICS, based on finding a GIS. We show that the prior state-of-the-art approaches yield integer-linear programming (ILP) models whose sizes grow exponentially with the problem size and $k$, while our GIS encoding only grows polynomially with the problem size and $k$. While the ILP approach can solve the GICS problem on networks of at most 494 nodes, the GIS-based method can handle networks of up to 21363 nodes; a $\sim 40\times$ improvement. The GIS-based method shows up to a $520\times$ improvement on the ILP-based method in terms of median solving time. For the majority of the instances that can be encoded and solved by both methods, the cardinality of the solution returned by the GIS-based method is less than $10\%$ larger than the cardinality of the solution found by the ILP method.

翻译：网络科学中的一个重要问题是在节点中寻找传感器的最优部署位置，以唯一地检测网络中的故障。该问题可建模为标识码集问题，由Karpovsky等人于1998年提出。标识码集问题旨在找到集合$S$的一个覆盖，使得覆盖中的元素为$S$中的每个元素定义唯一签名，并最小化覆盖的基数。本研究考虑广义标识码集问题，其中需为$S$中基数至多为$k$的每个子集（而非$S$的每个元素）找到唯一签名。布尔公式独立支持的概念由Chakraborty等人于2014年提出，通过识别一组变量（其真值赋值唯一确定其他变量的赋值）来加速命题模型计数。本研究引入独立支持的扩展版本——分组独立支持，并展示如何将广义标识码集问题归约为分组独立支持问题。随后，我们提出一种基于分组独立支持求解的新方法。研究表明，现有最优方法生成的整数线性规划模型规模随问题规模和$k$呈指数增长，而我们的分组独立支持编码仅随问题规模和$k$呈多项式增长。整数线性规划方法最多能解决494个节点网络上的广义标识码集问题，而基于分组独立支持的方法可处理多达21363个节点的网络，提升约40倍。基于分组独立支持的方法在求解时间中位数上较整数线性规划方法提升高达520倍。对于两种方法均可编码求解的大多数实例，基于分组独立支持方法返回的解的基数比整数线性规划方法找到的解的基数大不到10%。